You nearly got it. Look at it this way. Islam came to India, just as it did in other countries and spread until a strong Muslim community was formed. Many great and traditional Ulema were produced during this time, from Hadhrat Mujaddid Alf Thani to Shah Waliullah Dehlwi (Allah have mercy on them) and beyond. At some point during this time (the 1600s to be more precise), the British arrived and eventually, Islam began to diminish in the region until some traditional Ulema; who were inheritors of the same traditional legacy left behind by Shah Waliullah (rahmatullahialayh), decided to revive it once more through the establishment of a Madarsa at Deoband. They not only preserved and continued the spread and growth of traditional Islam through actively teaching it, but in addition, also fought against the Shirk and Bid'ah prevalent in the Subcontinent at the time. These Ulema were not a part of a new sect and neither did they claim to be. They were from the same line of Hanafi Ulema to which the likes of Hadhrat Imam Abu Hanifah, Imams Tahawi, Sarakhsi, Ibn Abideen, Shah Waliullah and numerous others (may Allah have mercy on them all) also belonged. The success of the Madarsa was such that soon, its influence had spread to the far reaches of the Subcontinent and under its supervision, numerous Madaaris were rendering their services of Deen to the extent that the vast majority of religious institutions in the region were operating under the supervision of Dar ul Uloom Deoband and its affiliated institutions like Mazaahir ul Uloom Sahaaranpur. On seeing this, certain other Ulema who also claimed to be Hanafi but supported the practices of Bid’ah which the Madarsa at Deoband opposed, rose up in opposition to the Ulema of the Madarsa at Deoband. These are the Ulema we now refer to as Barelwis. However, since these Ulema also claimed to be followers of the traditional Islamic legacy and the Hanafi School (which in my view is nothing but an empty claim of theirs and is not supported by facts), it became necessary to distinguish between the Hanafi Ulema who adhered to the view of the Ulema of Deoband (i.e. opposition to Bid’ah) and those Hanafi Ulema that did not (i.e. who adhered to the practices of Bid’ah). As a result, the supporters of Bid’ah began referring to the Ulema from the former group as Deobandis and themselves as Barelwis. Therefore, in reality, there is no such thing as a “Deobandi”. This word is just an unfortunate consequence of the ploys of the evil and corruption which existed in the Subcontinent at the time. Otherwise, the “Deobandis” are simply a continuation of the same Islam taught by the traditional Hanafi Ulema for the last 13-14 centuries. I know Barelwis will also claim the same and accuse us of being the deviants, but the facts do not support their claims. This is how I see it. Anyone is free to correct me.